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1)     in the diagram below, we have $\overline{QR}\parallel\overline{ST}$$PQ = 8$$QS = 6$, and $PT = 10.5$. Find $PR$.

[asy] size(5cm);pair P,Q,R,SS,T;P = (0,0);Q = (0.5,0.6);R = (0.6,0);SS = 1.75*Q;T = 1.75*R;draw (Q--R--T--SS--P--R);label(

 

2)    In the diagram below, we have $\overline{MN}\parallel\overline{PQ}$. We also know that $OM = 9$$OQ = 6$, and $PN = 25$. Find $OP$

[asy] size(5cm);pair O,M,NN,P,Q;M = (-0.3,0.5);NN = (0.6,0.5);P = -0.7*NN;Q = -0.7*M;label(

 

 

 

 

3)       In the diagram, angles $ABC$ and $ADE$ are right angles. If $AC = 35$$AE = 11$, and $BE = 10$, then what is $AD$?

[asy] pair A,B,C,D,EE;B = (0,0);A = (1,0);C = (0,0.7);EE = (0.4,0);D = foot(EE,A,C);draw(D--EE--A--C--B--E);draw(rightanglemark(A,B,C,1.5));draw(rightanglemark(EE,D,A,1.5));label(

 

 

4)        $E$ is a point on side $\overline{CD}$ of square $ABCD$ such that $DE=2$ and $EC=5.$ The extensions of $\overline {AD}$ and $\overline {BE}$ intersect at $F.$ Find $DF.$

[asy] size(5cm);pair A,B,C,D,EE,F;A=(0,0);B=(6,0);D=(0,6);C=B+D;EE=(C+2D)/3;F=3D/2;draw(B--C--D--F--B--A--D);label(

 

 

 

5)          In the diagram below, we have $\triangle MNO\sim\triangle NRO\sim\triangle QPO.$ What is $\angle MNR$ in degrees?

[asy] pair O,M,NN,P,Q,R;M = (-0.4,0.5);R=0.75M;NN = (0.3,0.5);P = -0.9*NN;Q = -0.9*M;label(

 

 

 

6)       Segments $\overline{AC}$ and $\overline{BD}$ intersect at $O$. We know that $\dfrac{OA}{OB}=\dfrac{OD}{OC}$. Which angles below are equal to $\angle CDB$?

(w) $\angle ADB$
(x) $\angle CAB$
(y) $\angle BCA$
(z) $\angle ABD+\angle ACD$
[asy] size(5cm);pair A,B,C,D,O;A = 2*dir(15);B = 4*dir(140);C = -3A;D = -0.75B;draw(A--B--C--D--A--C^^B--D);label(

Guest Oct 27, 2017
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you do not have to explain how you do it so if you want to give just the answer you can   smiley

Guest Oct 28, 2017

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